calculate the invariant vectors for a representation of a Lie algebra, calculate the invariant tensors for a tensor product representation of a Lie algebra

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LieAlgebras[Invariants] - calculate the invariant vectors for a representation of a Lie algebra, calculate the invariant tensors for a tensor product representation of a Lie algebra

Calling Sequences

Invariants(rho)

Invariants(rho, T)

Parameters

rho - a representation of a Lie algebra g on a vector space V

T - list of tensors on V defining a subspace of tensors invariant under the induced representation of rho

Description

•	Let rho: g -> gl(V) be a representation of a Lie algebra g on a vector space V. A vector y in V is an invariant vector for the representation rho if rho(x)(y) = 0 for all x in g.

•	The procedure Invariants(rho) returns a list of the invariant vectors for the representation rho.

•	The procedure Invariants(rho, T) returns a list of the invariant tensors for the induced representation of rho acting on the tensors T.

Examples

Example 1.

We define a 6 dimensional representation of sl2 and find the invariant vectors.

(2.1)

sl2 >

W1 >

(2.2)

sl2 >

(2.3)

We check this result using the ApplyRepresentation command.

W1 >

(2.4)

Example 2.

In this example we calculate the invariant (1, 1) tensors, the invariant (0, 2) symmetric tensors and the type (1, 2) invariant tensors for the adjoint representation of the Lie algebra [3, 2] in the Winternitz tables of Lie algebras. We begin by using the Retrieve command to obtain the the structure equations for this Lie algebra.

W1 >

(2.5)

V >

Alg1 >

V >

(2.6)

There are no vector invariants.

Alg1 >

(2.7)

V >

(2.8)

There is one 1-form invariant.

Alg1 >

(2.9)

V >

(2.10)

There is 1 invariant type (1, 1) tensor.

V >

(2.11)

V >

(2.12)

There is 1 invariant symmetric type (0, 2) tensor (but no invariant metrics).

V >

(2.13)

V >

(2.14)

There are 3 type (1, 2) invariant tensors.

V >

(2.15)

V >

(2.16)

We can check the validity of the these calculations in two steps. First we use the matrices for the representation rho2 to construct linear vector fields on the representation space V. This gives a vector field realization Gamma of our Lie algebra. The invariance of the tensors Inv1, Inv2, Inv3 means that the Lie derivatives of these tensors with respect to the vector fields in Gamma vanishes.

V >

A := [Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = 0, (1, 2) = 0, (1, 3) = 1, (2, 1) = 0, (2, 2) = 0, (2, 3) = 1, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0}), Matrix(3, 3, {(1, 1) = -1, (1, 2) = -1, (1, 3) = 0, (2, 1) = 0, (2, 2) = -1, (2, 3) = 0, (3, 1) = 0, (3, 2) = 0, (3, 3) = 0})]

(2.17)

Alg1 >

(2.18)

V >

(2.19)

Use the LieDerivative command to verify the invariance of the the tensors calculated by the Invariants command.

V >

$Matrix(1, 3, {(1, 1) = `0`*dx*D_x, (1, 2) = `0`*dx*D_x, (1, 3) = `0`*dx*D_x})$

(2.20)

V >

$Matrix(1, 3, {(1, 1) = `0`*dx^2, (1, 2) = `0`*dx^2, (1, 3) = `0`*dx^2})$

(2.21)

V >

$Matrix(3, 3, {(1, 1) = `0`*dx^2*D_x, (1, 2) = `0`*dx^2*D_x, (1, 3) = `0`*dx^2*D_x, (2, 1) = `0`*dx^2*D_x, (2, 2) = `0`*dx^2*D_x, (2, 3) = `0`*dx^2*D_x, (3, 1) = `0`*dx^2*D_x, (3, 2) = `0`*dx^2*D_x, (3, 3) = `0`*dx^2*D_x})$